I travel a fair bit and always have a carry on bag, (which always has a change of clothes, toiletries of the size one is allowed to carry on and plugs, cords, adaptors, etc) and a briefcase. My laptop travels in my briefcase, along with books, papers and my iPod.
At security, you will be required to remove the laptop from whatever bag it is in, in order to to be examined by the x-ray machines. Keep an eye on your possessions.
When travelling through airport security make sure your passport and ticket are tucked away securely; I always put mine in the inside breast pocket of my jacket and it is the very first thing I do, before removing my jacket and wallet to place them in the airport security x-ray conveyer belt. This is because I have known individuals who left them behind, beside an x-ray machine, which meant a nightmare later on.
Do not put anything valuable in your checked-in luggage.
Drink plenty of water before your flight, and, after you have gone through security, buy a few plastic bottles of water to drink on the flight. I don't advise consuming too much alcohol in the airport; for one thing, - even without alcohol - you get dehydrated on a long flight; for another, arriving in a strange - by which I mean, unknown - country ever so slightly intoxicated may not be a terribly good idea.
As Happybunny says, do not argue with (but be polite to) airport security personnel.
Meanwhile, good luck and enjoy your extended visit; Rome is brilliant, especially if you are there to study art and design.